Question

How do you write 1.2244444... as a fraction?

Answer 1

Hint: Now to write the number in decimal form we will first assume the number to be x. Now we will multiply the number by 100 to form our first equation. Now again we will multiply the number with 10 to form a second equation. Now we will subtract both the equations and then divide with the coefficient of x. Hence we have x in fractional form.

Complete step by step solution:
Now we know that the given number is a recurring decimal.
Now we want to convert the decimal into fraction. Hence we want to write the number in $\dfrac{p}{q}$ form where p and q are both integers.
Now consider the given number 1.224444…
Let x = 1.224444…
Now the decimal is recurring after 2 decimal places. Hence we will first multiply the number equation by 100.
Hence we get, $100x=122.4444............\left( 1 \right)$
Now again multiply the above number by 10. Hence we have,
$\Rightarrow 1000x=1224..444............\left( 2 \right)$
Now subtracting equation (1) from equation (2) we get,
$\Rightarrow 1000x-100x=1224.444...-122.444....$
Now on subtraction we get,
$\Rightarrow 900x=1102$
Now dividing the whole equation by 900 we get,
$\Rightarrow x=\dfrac{1102}{900}$
Now x is nothing but 1.224444…
Hence we have fractional representation of the given number as $\dfrac{1102}{900}$
Now on simplifying the fraction by dividing the numerator and denominator by 2 we get $\dfrac{551}{450}$

Note: Now note that all numbers cannot be written into fraction forms. The number which can be written into fraction form is called rational numbers. Now such numbers if written in decimal form are either terminating or recurring. Now since the given number is recurring we can easily write the number in decimal form to do so we first multiply the number with powers of 10 such that recurring starts after decimal point then we will again multiply with ${{10}^{n}}$ where n is the number of digits which are recurring. If the decimal is terminating then simply remove the decimal point and divide the number with ${{10}^{n}}$ where n is the number of digits after decimal point.